\(\int (c+d x)^3 (c^2-d^2 x^2)^{2/3} \, dx\) [249]

Optimal result

Integrand size = 24, antiderivative size = 707 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx=\frac {66}{91} c^3 x \left (c^2-d^2 x^2\right )^{2/3}-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}-\frac {33 c (13 c+5 d x) \left (c^2-d^2 x^2\right )^{5/3}}{520 d}-\frac {264 c^5 x}{91 \left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}-\frac {132 \sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{17/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{91 d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {88 \sqrt {2} 3^{3/4} c^{17/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{91 d^2 x \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}} \] Output:

66/91*c^3*x*(-d^2*x^2+c^2)^(2/3)-3/16*(d*x+c)^2*(-d^2*x^2+c^2)^(5/3)/d-33/ 
520*c*(5*d*x+13*c)*(-d^2*x^2+c^2)^(5/3)/d-264*c^5*x/(91*(1-3^(1/2))*c^(2/3 
)-91*(-d^2*x^2+c^2)^(1/3))-132/91*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*c^(17/ 
3)*(c^(2/3)-(-d^2*x^2+c^2)^(1/3))*((c^(4/3)+c^(2/3)*(-d^2*x^2+c^2)^(1/3)+( 
-d^2*x^2+c^2)^(2/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))^2)^(1/2)*E 
llipticE(((1+3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))/((1-3^(1/2))*c^(2/3)-( 
-d^2*x^2+c^2)^(1/3)),2*I-I*3^(1/2))/d^2/x/(-c^(2/3)*(c^(2/3)-(-d^2*x^2+c^2 
)^(1/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))^2)^(1/2)+88/91*2^(1/2) 
*3^(3/4)*c^(17/3)*(c^(2/3)-(-d^2*x^2+c^2)^(1/3))*((c^(4/3)+c^(2/3)*(-d^2*x 
^2+c^2)^(1/3)+(-d^2*x^2+c^2)^(2/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1 
/3))^2)^(1/2)*EllipticF(((1+3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))/((1-3^( 
1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3)),2*I-I*3^(1/2))/d^2/x/(-c^(2/3)*(c^(2/3 
)-(-d^2*x^2+c^2)^(1/3))/((1-3^(1/2))*c^(2/3)-(-d^2*x^2+c^2)^(1/3))^2)^(1/2 
)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 6.98 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.15 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx=\frac {\left (c^2-d^2 x^2\right )^{2/3} \left (-\frac {1053 c^4}{d}-720 c^3 x+858 c^2 d x^2+720 c d^2 x^3+195 d^3 x^4+\frac {1760 c^3 x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},\frac {d^2 x^2}{c^2}\right )}{\left (1-\frac {d^2 x^2}{c^2}\right )^{2/3}}\right )}{1040} \] Input:

Integrate[(c + d*x)^3*(c^2 - d^2*x^2)^(2/3),x]
 

Output:

((c^2 - d^2*x^2)^(2/3)*((-1053*c^4)/d - 720*c^3*x + 858*c^2*d*x^2 + 720*c* 
d^2*x^3 + 195*d^3*x^4 + (1760*c^3*x*Hypergeometric2F1[-2/3, 1/2, 3/2, (d^2 
*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^(2/3)))/1040
 

Rubi [A] (warning: unable to verify)

Time = 0.97 (sec) , antiderivative size = 792, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {497, 27, 497, 27, 455, 211, 233, 833, 760, 2418}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx\)

\(\Big \downarrow \) 497

\(\displaystyle -\frac {3 \int -\frac {22}{3} c d^2 (c+d x)^2 \left (c^2-d^2 x^2\right )^{2/3}dx}{16 d^2}-\frac {3 \left (c^2-d^2 x^2\right )^{5/3} (c+d x)^2}{16 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11}{8} c \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{2/3}dx-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

\(\Big \downarrow \) 497

\(\displaystyle \frac {11}{8} c \left (-\frac {3 \int -\frac {16}{3} c d^2 (c+d x) \left (c^2-d^2 x^2\right )^{2/3}dx}{13 d^2}-\frac {3 (c+d x) \left (c^2-d^2 x^2\right )^{5/3}}{13 d}\right )-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {11}{8} c \left (\frac {16}{13} c \int (c+d x) \left (c^2-d^2 x^2\right )^{2/3}dx-\frac {3 (c+d x) \left (c^2-d^2 x^2\right )^{5/3}}{13 d}\right )-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {11}{8} c \left (\frac {16}{13} c \left (c \int \left (c^2-d^2 x^2\right )^{2/3}dx-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\right )-\frac {3 (c+d x) \left (c^2-d^2 x^2\right )^{5/3}}{13 d}\right )-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {11}{8} c \left (\frac {16}{13} c \left (c \left (\frac {4}{7} c^2 \int \frac {1}{\sqrt [3]{c^2-d^2 x^2}}dx+\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\right )-\frac {3 (c+d x) \left (c^2-d^2 x^2\right )^{5/3}}{13 d}\right )-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {11}{8} c \left (\frac {16}{13} c \left (c \left (\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}-\frac {6 c^2 \sqrt {-d^2 x^2} \int \frac {\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}}{7 d^2 x}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\right )-\frac {3 (c+d x) \left (c^2-d^2 x^2\right )^{5/3}}{13 d}\right )-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {11}{8} c \left (\frac {16}{13} c \left (c \left (\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}-\frac {6 c^2 \sqrt {-d^2 x^2} \left (\left (1+\sqrt {3}\right ) c^{2/3} \int \frac {1}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}\right )}{7 d^2 x}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\right )-\frac {3 (c+d x) \left (c^2-d^2 x^2\right )^{5/3}}{13 d}\right )-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {11}{8} c \left (\frac {16}{13} c \left (c \left (\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}-\frac {6 c^2 \sqrt {-d^2 x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\sqrt {-d^2 x^2}}d\sqrt [3]{c^2-d^2 x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}\right )}{7 d^2 x}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\right )-\frac {3 (c+d x) \left (c^2-d^2 x^2\right )^{5/3}}{13 d}\right )-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {11}{8} c \left (\frac {16}{13} c \left (c \left (\frac {3}{7} x \left (c^2-d^2 x^2\right )^{2/3}-\frac {6 c^2 \sqrt {-d^2 x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right ) \sqrt {\frac {c^{4/3}+\left (c^2-d^2 x^2\right )^{2/3}+c^{2/3} \sqrt [3]{c^2-d^2 x^2}}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-d^2 x^2} \sqrt {-\frac {c^{2/3} \left (c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )}{\left (\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}\right )^2}}}-\frac {2 \sqrt {-d^2 x^2}}{\left (1-\sqrt {3}\right ) c^{2/3}-\sqrt [3]{c^2-d^2 x^2}}\right )}{7 d^2 x}\right )-\frac {3 \left (c^2-d^2 x^2\right )^{5/3}}{10 d}\right )-\frac {3 (c+d x) \left (c^2-d^2 x^2\right )^{5/3}}{13 d}\right )-\frac {3 (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/3}}{16 d}\)

Input:

Int[(c + d*x)^3*(c^2 - d^2*x^2)^(2/3),x]
 

Output:

(-3*(c + d*x)^2*(c^2 - d^2*x^2)^(5/3))/(16*d) + (11*c*((-3*(c + d*x)*(c^2 
- d^2*x^2)^(5/3))/(13*d) + (16*c*((-3*(c^2 - d^2*x^2)^(5/3))/(10*d) + c*(( 
3*x*(c^2 - d^2*x^2)^(2/3))/7 - (6*c^2*Sqrt[-(d^2*x^2)]*((-2*Sqrt[-(d^2*x^2 
)])/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3)) + (3^(1/4)*Sqrt[2 + Sq 
rt[3]]*c^(2/3)*(c^(2/3) - (c^2 - d^2*x^2)^(1/3))*Sqrt[(c^(4/3) + c^(2/3)*( 
c^2 - d^2*x^2)^(1/3) + (c^2 - d^2*x^2)^(2/3))/((1 - Sqrt[3])*c^(2/3) - (c^ 
2 - d^2*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*c^(2/3) - (c^2 - d^ 
2*x^2)^(1/3))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))], -7 + 4*Sqr 
t[3]])/(Sqrt[-(d^2*x^2)]*Sqrt[-((c^(2/3)*(c^(2/3) - (c^2 - d^2*x^2)^(1/3)) 
)/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[ 
3]]*(1 + Sqrt[3])*c^(2/3)*(c^(2/3) - (c^2 - d^2*x^2)^(1/3))*Sqrt[(c^(4/3) 
+ c^(2/3)*(c^2 - d^2*x^2)^(1/3) + (c^2 - d^2*x^2)^(2/3))/((1 - Sqrt[3])*c^ 
(2/3) - (c^2 - d^2*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*c^(2/3) 
- (c^2 - d^2*x^2)^(1/3))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))], 
 -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-(d^2*x^2)]*Sqrt[-((c^(2/3)*(c^(2/3) - (c^ 
2 - d^2*x^2)^(1/3)))/((1 - Sqrt[3])*c^(2/3) - (c^2 - d^2*x^2)^(1/3))^2)])) 
)/(7*d^2*x))))/13))/8
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) 
   Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b 
}, x]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 497

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b 
*(n + 2*p + 1))   Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 
 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n 
, p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p 
+ 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 833

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] 
], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r)   Int[1/Sqrt[a + b*x 
^3], x], x] + Simp[1/r   Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x 
]] /; FreeQ[{a, b}, x] && NegQ[a]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \left (d x +c \right )^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}d x\]

Input:

int((d*x+c)^3*(-d^2*x^2+c^2)^(2/3),x)
 

Output:

int((d*x+c)^3*(-d^2*x^2+c^2)^(2/3),x)
 

Fricas [F]

\[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx=\int { {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{3} \,d x } \] Input:

integrate((d*x+c)^3*(-d^2*x^2+c^2)^(2/3),x, algorithm="fricas")
 

Output:

integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*(-d^2*x^2 + c^2)^(2/3), 
 x)
 

Sympy [A] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.28 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx=c^{\frac {13}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )} + c^{\frac {7}{3}} d^{2} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )} + 3 c^{2} d \left (\begin {cases} \frac {x^{2} \left (c^{2}\right )^{\frac {2}{3}}}{2} & \text {for}\: d^{2} = 0 \\- \frac {3 \left (c^{2} - d^{2} x^{2}\right )^{\frac {5}{3}}}{10 d^{2}} & \text {otherwise} \end {cases}\right ) + d^{3} \left (\begin {cases} - \frac {9 c^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {2}{3}}}{80 d^{4}} - \frac {3 c^{2} x^{2} \left (c^{2} - d^{2} x^{2}\right )^{\frac {2}{3}}}{40 d^{2}} + \frac {3 x^{4} \left (c^{2} - d^{2} x^{2}\right )^{\frac {2}{3}}}{16} & \text {for}\: d \neq 0 \\\frac {x^{4} \left (c^{2}\right )^{\frac {2}{3}}}{4} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((d*x+c)**3*(-d**2*x**2+c**2)**(2/3),x)
 

Output:

c**(13/3)*x*hyper((-2/3, 1/2), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c**2) + 
 c**(7/3)*d**2*x**3*hyper((-2/3, 3/2), (5/2,), d**2*x**2*exp_polar(2*I*pi) 
/c**2) + 3*c**2*d*Piecewise((x**2*(c**2)**(2/3)/2, Eq(d**2, 0)), (-3*(c**2 
 - d**2*x**2)**(5/3)/(10*d**2), True)) + d**3*Piecewise((-9*c**4*(c**2 - d 
**2*x**2)**(2/3)/(80*d**4) - 3*c**2*x**2*(c**2 - d**2*x**2)**(2/3)/(40*d** 
2) + 3*x**4*(c**2 - d**2*x**2)**(2/3)/16, Ne(d, 0)), (x**4*(c**2)**(2/3)/4 
, True))
 

Maxima [F]

\[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx=\int { {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{3} \,d x } \] Input:

integrate((d*x+c)^3*(-d^2*x^2+c^2)^(2/3),x, algorithm="maxima")
 

Output:

integrate((-d^2*x^2 + c^2)^(2/3)*(d*x + c)^3, x)
 

Giac [F]

\[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx=\int { {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{3} \,d x } \] Input:

integrate((d*x+c)^3*(-d^2*x^2+c^2)^(2/3),x, algorithm="giac")
 

Output:

integrate((-d^2*x^2 + c^2)^(2/3)*(d*x + c)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx=\int {\left (c^2-d^2\,x^2\right )}^{2/3}\,{\left (c+d\,x\right )}^3 \,d x \] Input:

int((c^2 - d^2*x^2)^(2/3)*(c + d*x)^3,x)
 

Output:

int((c^2 - d^2*x^2)^(2/3)*(c + d*x)^3, x)
 

Reduce [F]

\[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{2/3} \, dx=\frac {-7371 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} c^{4}+240 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} c^{3} d x +6006 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} c^{2} d^{2} x^{2}+5040 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} c \,d^{3} x^{3}+1365 \left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}} d^{4} x^{4}+7040 \left (\int \frac {1}{\left (-d^{2} x^{2}+c^{2}\right )^{\frac {1}{3}}}d x \right ) c^{5} d}{7280 d} \] Input:

int((d*x+c)^3*(-d^2*x^2+c^2)^(2/3),x)
 

Output:

( - 7371*(c**2 - d**2*x**2)**(2/3)*c**4 + 240*(c**2 - d**2*x**2)**(2/3)*c* 
*3*d*x + 6006*(c**2 - d**2*x**2)**(2/3)*c**2*d**2*x**2 + 5040*(c**2 - d**2 
*x**2)**(2/3)*c*d**3*x**3 + 1365*(c**2 - d**2*x**2)**(2/3)*d**4*x**4 + 704 
0*int((c**2 - d**2*x**2)**(2/3)/(c**2 - d**2*x**2),x)*c**5*d)/(7280*d)